Implications for Hedging of the choice of driving process for one-factor Markov-functional models

Transcription

1 Implcatons for Hedgng of the choce of drvng process for one-factor Marov-functonal models Joanne E. Kennedy and Duy Pham Department of Statstcs, Unversty of Warwc December 16, 211 Abstract In ths paper, we study the mplcatons for hedgng Bermudan swaptons of the choce of the nstantaneous volatlty for the drvng Marov process of the one-dmensonal swap Marov-functonal model. We fnd that there s a strong evdence n favour of what we term parametrzaton by tme as opposed to parametrzaton by expry. We further propose a new parametrzaton by tme for the drvng process whch taes as nputs nto the model the maret correlatons of relevant swap rates. We show that the new drvng process enables a very effectve vega-delta hedge wth a much more stable gamma profle for the hedgng portfolo compared wth the exstng ones. Key words: one-dmensonal swap Marov-functonal model, Bermudan swapton, correlaton, hedgng, vega, gamma, parametrzaton by tme and by expry. Contents 1 Introducton 2 2 Notatons and prelmnares 3 3 Prcng Bermudan swaptons under the one-dmensonal swap Marov-functonal model The one-dmensonal swap Marov-functonal model Parametrzatons by tme and by expry An alternatve parametrzaton of tme One step covarance Weghted covarance Vegas The vega computaton under the swap Marov-functonal model The Bermudan swapton s vegas under the HW and MR models The Bermudan swapton s vegas under the one step and weghted covarance models The net maret vegas for dfferent parameterzatons

2 5 A hedgng result A hedgng portfolo for the HW and MR models Vega hedge Delta hedge The gammas of the HW and MR hedgng portfolos A hedgng portfolo for the one step and weghted covarance models Vega hedge Delta hedge The gammas of the one step and weghted covarance hedgng portfolos Conclusons 37 References 38 A Estmatng the maret mpled covarance/correlaton structure 38 A.1 Approxmatng the termnal correlatons, a global ft approach A.2 Approxmatng the covarances, a local ft approach Introducton The problem of prcng and hedgng a Bermudan swapton has been of great practcal concern n the fxed ncome quanttatve research. The product tself s among the most common exotc nterest rate dervatves. However, opnons dffer as to what consttutes an effectve modelng framewor for prcng and hedgng Bermudan swaptons. One of the bggest debates s whether t s necessary to use a mult-factor model. A good summary of the current lterature on ths topc s gven n (Petersz & Pelsser, 21. In (Petersz & Pelsser, 21, the authors carry out a comparson of the hedgng performance of a sngle factor Marov-functonal model and mult-factor maret models n relaton to Bermudan swaptons and ther fndngs support the clam that f a sngle factor Marov-functonal model s approprately calbrated to termnal correlatons of swap rates that are relevant to the Bermudan swapton then the hedgng performance of both the mult-factor and sngle factor models are comparable. In ths paper, we restrct attenton to the prcng and hedgng of Bermudan swaptons wthn the context of a one factor Marov-functonal model drven by a Gaussan process. The contrbuton we mae here s to study the mplcatons for hedgng of the choce of the nstantaneous volatlty for the drvng Marov process. Ths s a topc whch seems to have receved lttle attenton n the lterature for one factor Marov-functonal models or equvalently for one factor separable maret models (see (Bennett & Kennedy, 25 and (Petersz, Pelsser, & Regenmortel, 24. One popular choce s to tae a Gaussan process wth exponental nstantaneous volatlty, referred to as the mean reverson process (MR, as s done n (Petersz & Pelsser, 21. We begn our nvestgaton by comparng ths canddate wth one based on the Hull-Whte short-rate model, referred to as the Hull-Whte process (HW whch was frst ntroduced n (Bennett & Kennedy, 25. For these two canddate processes the vega profles of a Bermudan swapton under the swap Marov-functonal model turn out to have some ey dfferences (see also (Pertursson, 28 for a comparson of ther vega profles under dfferent maret scenaros. These dfferences can be lned bac to the dfference n nature of the two parametrzatons for the drvng process. The mean reverson process (MR s an example of what we term parametrzaton by expry. Here 2

3 the auto-correlatons of the drvng process are chosen at the outset and controlled by parameters whch are user nputs. As such the changes n the correlatons of swap rates at ther settng dates relevant to the prcng of a Bermudan are not hedged. In contrast, the Hull-Whte process (HW s an example of parametrzaton by tme. In ths type of parametrzaton, the auto-correlatons of the drvng process are lned explctly to maret mpled volatltes and t s ths feature whch allows the possblty of hedgng aganst moves n maret correlatons of relevant swap rates. Based on the nsght ganed by our study of the MR and HW processes, we propose a new parametrzaton by tme for the drvng process. Ths new parametrzaton taes as nputs nto the model the maret correlatons of relevant swap rates. These maret correlatons are estmated va a full ran LIBOR maret model usng a two-step procedure nvolvng a global and local ft to the swapton matrx. Ths new parametrzaton has a vega response spread over the swapton matrx but nterestngly the total vega for each expry (row of the swapton matrx s approxmately the same as for the HW model. We gve an explanaton for why ths s the case. The dfferent vega profles of the parametrzatons by expry and by tme have a drect consequence for hedgng. We fnd that when the drvng process s parameterzed by tme the total gamma (sum of all gammas of a vega-delta neutral portfolo for a Bermudan swapton s stablzed. In contrast, t s not possble to control the total gamma for ths portfolo wth the vega profle assocated wth parametrzaton by expry. We further fnd that the proposed parametrzaton by tme for the drvng process wth a vega response spread over the swapton matrx leads to a more stable parallel gamma profle (sum of each row of the gamma matrx than that of the HW process. The paper s organzed as follows. In secton 2, we revew the prelmnares and set up the notatons. In secton 3, we frst descrbe the one-dmensonal swap Marov-functonal model and analyze the dfference between parametrzatons by expry and by tme. After that, we construct a new parametrzaton by tme for the drvng process. In secton 4, we compute the vegas of a Bermudan and analyze them theoretcally. A hedgng result wth an emphass on the gamma rss wll be addressed n secton 5. Secton 6 concludes the paper. 2 Notatons and prelmnares Consder a general tenor structure = T < T 1 < < T n+1, where α = +1 are the accrual factors for =,..., n. Let D tt denote the tme-t value of a zero-coupon dscount bond that matures at tme T. We denote by L the forward LIBOR that sets (expres at and settles (matures at +1. Forward LIBORs and dscount bonds can be lned va the relaton L t = D t D tt+1 α D tt+1, t, (1 for =,..., n. We denote by y,j the forward swap rate of an nterest rate swap wth settng dates, +1,..., +j 1 and settlement dates +1, +2,..., +j. Smlar to forward LIBORs, forward swap rates can also be wrtten n terms of dscount bonds y,j t = D t D tt+j +j 1 = α D tt+1, t, (2 3

4 for =,..., n. It s clear that y,1 concdes wth L. For each swap rate y,j, we further ntroduce the correspondng at the money (ATM Blac mpled volatlty σ,j. The type of Bermudan swapton we consder n ths paper s the co-termnal verson, as opposed to other non-standard types of Bermudan swapton. The holder of a co-termnal Bermudan swapton has the rght, on any of the swap exercse dates to enter the remanng swap whch ends at the pre-determned termnal date T n+1. The underlyng swap at conssts of a number of coupons that set at T j and settle at T j+1 for j =,..., n. We further denote the notonal amount by N and the stre by K. Suppose that the Bermudan swapton s to be exercsed at tme. In case of a pay fxed type, the holder wll then receve the correspondng coupons from the underlyng swap,.e. at T j+1 for each j =,..., n he or she wll receve the floatng leg Nα j L j T j and pay the fxed leg Nα j K. In case of a receve fxed type, the holder wll receve the fxed legs n exchange for the floatng legs. Although the coupons depend on the values of the LIBORs at ther settng dates, the exercse value of the underlyng swap at each exercse date depends on the correspondng co-termnal swap rate at ts settng date y,n+1. For a pay fxed Bermudan, the holder wll only exercse at tme f y,n+1 s above the stre level K. Nevertheless, even when the mmedate exercse value s postve, the holder can decde to hold on to the swapton n vew of a more favourable co-termnal swap rate y j,n+1 j for j >. It was noted n (Petersz & Pelsser, 21 that although the jont dstrbuton of the random varables {y j,n+1 j ; j =,..., n; = 1,..., n} fully determnes the prce of a Bermudan swapton, the man contrbuton (up to frst order approxmaton actually comes from the jont dstrbuton of the co-termnal swap rates at ther settng dates {y,n+1 ; = 1,..., n} (see also (Pterbarg, 24. Ths s why we are nterested n ther correlaton structure. 3 Prcng Bermudan swaptons under the one-dmensonal swap Marov-functonal model The defnng characterstc of the standard Marov-functonal model (MF s that dscount bond prces are assumed to be at any tme functons of some low-dmensonal (usually one or two Marov process x, whch s Marovan n some specfed martngale measure. The exact forms are only determned at the exercse dates,.e. D T T j (x T for j n, snce ths s all that s typcally needed n practce. Dependng on the applcaton, the functonal forms are derved numercally from relevant maret prces and the martngale property necessary to mantan the arbtrage-free property of the model. Note that the functonal forms of the dscount bonds mplctly mply the functonal forms of all forward swap/libor rates and vce versa. Gven the functonal forms, the condtonal expected value under the specfed martngale measure of a payoff at any exercse date can be derved numercally. Hence, the value of an exotc product can be calculated by bacward nducton on a grd. Here, we restrct attenton to the development of the one-dmensonal swap Marov-functonal model (SMF for the prcng and hedgng of Bermudan swaptons. Ths secton starts by revewng the one-dmensonal SMF model and ts current choces of drvng Marov process. We then propose an alternatve choce whch s more sutable for our current applcaton. 3.1 The one-dmensonal swap Marov-functonal model In the one-dmensonal SMF model, the functonal forms of the dscount bonds are chosen so that accurate calbraton to the maret prces of the co-termnal vanlla swaptons s acheved. 4

5 We assume that these maret prces are gven by the Blac s formula wth the correspondng cotermnal mpled volatltes { σ,n+1 } =1,...,n. The freedom to specfy the law of x allows the model to capture well some features of the real maret relevant to the exotc products. For a Bermudan swapton, those features are the correlatons of the co-termnal forward swap rates at ther settng dates as we dscussed n secton 2. In our model, we choose to wor wth the termnal measure S n+1 whch taes the termnal dscount bond D Tn+1 as the numerare. Detals of the mplementaton of the SMF model under the termnal measure can be found n (Hunt, Kennedy, & Pelsser, 2 and (Hunt & Kennedy, 24. We assume the drvng process x s a Gaussan process satsfyng x t := t σ(udw u, where W denotes a standard Brownan moton under S n+1 and σ( s a determnstc functon of tme. For the mplementaton of the model, we only need to specfy the law of x at each exercse date for = 1,..., n. An mportant result that was observed n (Bennett & Kennedy, 25 s the approxmate lnear relatonshp between the logarthms of the co-termnal forward swap rates and x ln y,n+1 t γ }{{} x t + ηt }{{}. (3 constant determnstc Consequently, the jont dstrbutons of the log of the co-termnal forward swap rates can be captured va our choce of x snce correlaton s nvarant under the lnear transformaton Corr(x T, x Tj = ξmn(t,t j ξ max(t,t j Corr mo (ln y,n+1, ln y j,n+1 j T j, where ξ T := Var(x T = σ2 (tdt and the superscrpt mo denotes model quanttes. We further note that by matchng the model to the Blac s formula for the co-termnal vanlla swaptons, we have the followng approxmaton n the termnal measure (exact n the assocated swapton measure Var mo (ln y,n+1 σ 2,n+1. (4 Hence, once x s chosen the γ s are mplctly determned and from (3 and (4 we have γ 2 ξ T σ 2,n+1. (5 Note that each γ s matched specfcally to the assocated co-termnal swap rate y,n+1 and once the model s calbrated t stays constant from today tll expry. In that sense, the γ s are expry-dependent quanttes. We now present n the followng two current canddates for x before explorng other choces n the later sectons. Current canddates: MR: The frst choce s referred to as the mean reverson (MR drvng process wth σ(t = e at, where a > s the mean reverson parameter. It follows that one can wrte the varance of x at each exercse date as ξ T = T e 2at dt = 1 2a (e2a 1. 5

6 For ths choce of parametrzaton, one can see that any changes n the maret mpled volatltes wll not nfluence x and ts auto-correlatons once we fx the parameter a. However, the expry-dependent quanttes γ s may change as we can see from (5. In that sense, the MR process s an example of parametrzaton by expry. HW: An alternatve choce of x s motvated by consderng the Hull-Whte short-rate model whch was frst ntroduced n (Bennett & Kennedy, 25. We refer to t as the Hull-Whte (HW process. For each = 1,..., n, we have the followng specfcaton for the HW process ξ T = ( 2 T n+1 (1 + α y,n+1 σ,n+1 T 2, (6 (ψ Tn+1 ψ T where ψ T = 1 a (1 e a, a >. In contrast to the MR process, any changes n the maret cotermnal mpled volatltes wll have an mmedate effect on x and ts auto-correlatons. From the lnear approxmaton n (3, we see that the nstantaneous volatltes of the co-termnal swap rates wll be altered n certan tme perods. On the other hand, the expry-dependent quanttes γ s wll stay the same as we can see from substtuton of the expresson (6 nto (5. In that sense, the HW process s an example of parametrzaton by tme. For both the above model parametrzatons, an ncrease n the mpled volatlty of one of the cotermnal swap rates tends to ncrease the value of a Bermudan. Ths s not surprsng as the value of the assocated vanlla swapton has ncreased. But the optonalty of a Bermudan provdes extra value n addton to the value of the underlyng vanlla optons. Ths extra value s hghly dependent on the correlatons between the co-termnal swap rates at ther settng dates and for the above two models these correlatons behave very dfferently n response to changes n the co-termnal mpled volatltes. Ths leads to very dfferent hedgng profles as we shall see n the later sectons. In the next secton we nvestgate the essental dfference n nature between the two parametrzatons by consderng the underlyng LIBORs. 3.2 Parametrzatons by tme and by expry In the prevous subsecton, we dscussed the dea of parametrzatons by expry and by tme n terms of the responses of the nstantaneous volatltes of the co-termnal swap rates to a shft n the mpled volatltes. Here we explore how ths dea carres over to the LIBORs as they are the basc buldng blocs of any nterest rate model. For all choces of x, the lnear approxmaton n (3 mples that the nstantaneous volatlty of the log of the co-termnal forward swap rate y,n+1 t s approxmately γ σ(t under the termnal measure. In order to gan nsght nto the effect of shftng the mpled volatltes, we mae the smplfyng assumpton that each log-libor ln L has a postve and determnstc volatlty functon σ (t, t. Under ths assumpton, we can use the approxmaton descrbed n appendx A. In a one factor model nstead of the mult-factor settng n appendx A, the nstantaneous volatlty of the log of the co-termnal forward swap rate y,n+1 can be lned to the nstantaneous volatltes of the log-libors by the followng approxmaton γ σ(t n = ζ,n+1 (σ (t, (7 6

7 where {ζ,n+1 (} =,...,n are constant emprcal weghts that depend on the ntal dscount curve. Ths can be seen from SDE (35 n appendx A.2. Snce the last LIBOR L n = y n,1, we have that σ n (t γ n σ(t. Usng the derved form for σ n (t, we can deduce σ n 1 (t by consderng the approxmaton n (7 for γ n 1 σ(t where we let σ n 1 (t σ n 1 σ(t, σ n 1 := γ n 1 ζn n 1,2 (γ n ζ n 1,2 n 1 (. We assume σ n 1 > so that σ n 1 ( wll also be postve as we assumed earler. Inductvely, assume we have that σ (t σ σ(t where σ s a postve constant for each = + 1,..., n. When = n, σ n s the same as γ n. We can then derve σ (t by consderng the approxmaton n (7 for γ σ(t. We rewrte (7 n the followng form γ σ(t ζ,n+1 (σ (t + σ (t ( γ n n =+1 =+1 ζ,n+1 ζ,n+1 ( Ths agan reduces σ (t approxmately to the form σ σ(t where σ := γ n =+1 ζ,n+1 ζ,n+1 ( ζ,n+1 (σ σ(t (σ σ(t. (σ. (8 Ths concludes that σ (t σ σ(t for all = 1,..., n where each constant σ can be derved nductvely by (8 and s assumed to be postve. One can see that each σ depends on {γ } =,...,n and the emprcal weghts {ζ,n+1 (} =,...,n, {ζ +1,n (} =+1,...,n,..., {ζ n 1,2 (} =n 1,n. Snce these emprcal weghts do not depend on the mpled volatltes, we wll safely gnore ther nvolvements n the next dscusson. We now analyze how shftng the mpled volatltes affects the nstantaneous volatlty functons {σ ( } =1,...,n of the log-libors for each choce of x. Parametrzaton by expry: For the MR process, by the approxmaton n (5 we have that γ σ,n+1 T e 2a 1 2a = 1,..., n. (9 Suppose we want to bump the co-termnal mpled volatlty σ,n+1 and eep the rest the same. It s clear that the nstantaneous volatlty σ(t = e at of the MR process wll not be affected. We observe other effects and summarze them below: γ j for j are unchanged as we can see from (9. Ths then follows from (8 that the constants {σ j } j=+1,...,n and hence {σ j ( } j=+1,...,n also reman unchanged. γ wll ncrease drectly as a result of (9. From (8, we see that σ and hence σ ( wll ncrease as {σ j } j=+1,...,n are unchanged. 7

8 Snce γ 1 and {σ j } j=+1,...,n stay the same but σ ncreases, agan we can see from (8 that σ 1 and hence σ 1 ( wll decrease. The effects on {σ } =1,..., 2 and hence {σ ( } =1,..., 2 wll be qute small. Ths s because the ncrease n σ and decrease n σ 1 tend to cancel each other out n the sum n (8 when we consder σ for < 1. Note that all the above effects are (global from today tll expry (llustrated n fgure 3.1. In that sense, the nstantaneous volatltes of the log-libors are clearly parameterzed by expry. Instantaneous volatlty σ n ( Parametrzaton by expry σ ( σ 1 ( σ 1 ( T T n Tme Fgure 3.1: Global effect of bumpng σ,n+1 on the nstantaneous volatlty functons of the log- LIBORs. The dots represent a very small effect. Parametrzaton by tme: For the HW process where ξ T s gven by (6, t can be seen from (5 that the γ s are ndependent of the mpled volatltes γ (1 + α y,n+1 (ψ Tn+1 ψ T T n+1, ψ T = 1 a (1 e a, a >. (1 Although the nstantaneous volatlty functon σ( of x s not defned explctly, we now t exsts such that σ2 (tdt = ξ T s gven by (6. We now bump the co-termnal mpled volatlty σ,n+1 and eep the rest unchanged. It s clear from (1 that γ j wll stay the same for j = 1,..., n. Hence, t follows from (8 that the constants σ j wll also reman unchanged for all j. The only effect of the bump s on the functon σ( (see (6. One can see that the varance of x at s shfted but those at the other exercse dates reman unchanged. Consequently, we have that the only effect on x s 8

9 the followng ξ T ξ T 1 = T 1 σ 2 (tdt ncreases, ξ T+1 ξ T = T+1 σ 2 (tdt decreases. The above effect mples that on average the nstantaneous volatlty functon σ( of x s ncreased durng the tme perod ( 1, ] but s decreased durng the next one (, +1 ]. Snce σ j (t σ j σ(t s only defned durng the correspondng LIBOR s lfe,.e. t [, T j ], the effect on σ( only carres over to {σ j ( } j=,...,n. It s then clear that on average the collecton of nstantaneous volatltes {σ j ( } j=,...,n wll ncrease and decrease durng the two consecutve tme ntervals ( 1, ] and (, +1 ] respectvely (fgure 3.2. For the last cotermnal mpled volatlty σ n,1, the equvalent effect s that σ n ( wll only ncrease durng the perod (T n 1, T n ]. Note that the above effects are local as the nstantaneous volatltes of the log-libors are only shoced locally for some partcular tme perods n response to the movement of the correspondng mpled volatlty. In that sense, the nstantaneous volatltes of the log-libors are clearly parameterzed by tme. Parametrzaton by tme Instantaneous volatlty σ n ( σ +1 ( σ ( σ 1 ( Tme T T n Fgure 3.2: Local effects of bumpng the σ,n+1 on the nstantaneous volatlty functons of the log-libors. The dfference n parametrzatons mentoned above has a fundamental effect on the hedgng of a Bermudan swapton. Specfcally, the global and local effects of the parametrzatons by expry and by tme nfluence the correlatons of the forward LIBORs and the co-termnal forward swap rates n very dfferent ways. Ths fact, n turn, leads to very dfferent hedgng behavours of correlatondependent products le the Bermudan swapton. We wll nvestgate further the dfference n ther vega profles n secton 4.2. Snce the parametrzaton by tme outperforms the other type as we explore later n secton 5, we wll next propose an alternatve for ths parametrzaton. 9

10 3.3 An alternatve parametrzaton of tme We recall that the correlaton of the MR process s fxed from the outset whle the HW specfcaton lns the correlaton structure of the model/co-termnal swap rates explctly to the maret mpled volatltes. However at each exercse date, the HW process only taes nto account the cotermnal mpled volatlty σ,n+1. In ths secton, we explore alternatve ways to specfy the x process whch ln the model s correlaton structure to mpled volatltes of dfferent tenors (see table 3.1. Tenor Expry σ 1, σ 2, σ 3, σ 4, σ 5, σ 6, σ 7, σ 8, σ 9, σ 1, σ 1,1 σ 1,2 σ 1,3 σ 1,4 σ 1,5 σ 1,6 σ 1,7 σ 1,8 σ 1,9 σ 1,1 σ 2,1 σ 2,2 σ 2,3 σ 2,4 σ 2,5 σ 2,6 σ 2,7 σ 2,8 σ 2,9... σ 3,1 σ 3,2 σ 3,3 σ 3,4 σ 3,5 σ 3,6 σ 3,7 σ 3, σ 4,1 σ 4,2 σ 4,3 σ 4,4 σ 4,5 σ 4,6 σ 4, σ 5,1 σ 5,2 σ 5,3 σ 5,4 σ 5,5 σ 5, σ 6,1 σ 6,2 σ 6,3 σ 6,4 σ 6, σ 7,1 σ 7,2 σ 7,3 σ 7, σ 8,1 σ 8,2 σ 8, σ 9,1 σ 9, σ 1, Table 3.1: Maret data from the swapton matrx to be ncorporated nto the drvng process x for a 11 years annual Bermudan swapton. HW s approach (left, alternatve approach (rght One step covarance One way to vew a Bermudan swapton s as the rght to choose between the assocated European swaptons. In settng up a model, one mght choose to try to capture the correlatons between the co-termnal swap rates at ther settng dates. These are the correlatons that matter when prcng a Bermudan swapton. In a one factor model, we cannot capture all these correlatons. One choce s to consder the one step correlatons,.e. the correlaton of y,n+1 wth ts nearest neghbour y +1,n +1 for each. Note that we wll wor wth the log of the swap rates because t allows for a drect connecton wth the drvng process x as we shall see n the model s setup. In the approach adopted here, we estmate the one step covarances Cov(ln y,n+1, ln y +1,n +1 for = 1,..., n 1 usng the swapton matrx from the maret. Ths s a two-step procedure. The frst step s to approxmate the correlatons of the log-libors at each exercse date by a global ft to the swapton matrx. The second step s to deduce the correspondng covarances of the log-libors by performng a local ft to each row of the swapton matrx and usng the correlatons from the frst step. We then use these covarances to derve the requred one step covarances (see equaton (36. In fact, we only need Cov(ln y,n+1, ln y +1,n as Cov(ln y,n+1, ln y j,n+1 j T j Cov(ln y,n+1, ln y j,n+1 j, for < j n (see appendx A.2. Detals of the whole approxmaton procedure can be found n appendx A. In what follows we wll use the superscrpt ma to denote quanttes estmated from the maret. Model s setup: Once we have estmated the one step covarances from the maret, we set up the model as follows. Recall the lnear approxmaton under the termnal measure n (Bennett & 1

11 Kennedy, 25, for < n γ x t + ηt ln y,n+1 t Corr(x T, x T+1 Corr mo (ln y,n+1, ln y +1,n +1 ξ T Cov mo (ln y,n+1 T, ln y +1,n. (11 ξ T+1 Var mo (ln y,n+1 Var mo (ln y +1,n +1 As for each = 1,..., n Var mo (ln y,n+1 can be nferred from the correspondng Blac mpled volatlty σ,n+1, we can now ncorporate the one step covarances nto the model as descrbed below. Wthout loss of generalty, fx ξ Tn = σ 2 n,1 T n. At T n 1, by nowledge of Cov ma (ln y n 1,2 T n 1, ln y n,1 T n and hence Corr ma (ln y n 1,2 T n 1, ln y n,1 T n = Cov ma (ln y n 1,2 T n 1,ln y n,1 Tn σ n 1,2 Tn 1 σ n,1 Tn that we have estmated from the maret, we can recover ξ Tn 1 by fxng ξ Tn 1 where we use the relaton n (11. = Corr ma (ln y n 1,2 T ξ n 1, ln y n,1 T n Tn ξ Tn 1 = Corr ma (ln y n 1,2 T n 1, ln y n,1 T n ξ Tn, Inductvely, assume we are at and have derved ξ Tj for j > from the prevous steps. By the approxmaton for Corr ma (ln y,n+1, ln y +1,n +1 from the maret and the nowledge of ξ T+1, we can fx ξ T where we agan use (11. ξ T = Corr ma (ln y,n+1, ln y +1,n +1 ξ T+1, We have now fxed ξ T for = 1,..., n and the SMF model can be mplemented on the grd. The above model s an example of parametrzaton by tme and ts overall vega profle (n terms of a Bermudan swapton has a close connecton to that of the HW model as we shall see n secton 4. As mpled volatltes change, the mpled correlatons n the maret change. The one step covarance model attempts to hedge ths rs but wth the focus just on the one step covarance wth the next co-termnal swap rate. Clearly, we are gnorng some mportant maret nformaton. Ths s exactly the motvaton for our next proposed model Weghted covarance We propose another alternatve choce of x based on the followng ntuton. In order to mae a decson whether to exercse the opton at, the one step correlaton Corr(ln y,n+1, ln y +1,n +1 11

12 wll be the most mportant as +1 s mmedately after. The correlatons wth the later rates are less sgnfcant snce we can wat untl later to decde. For each = 1,..., n 1, n addton to the one step covarance we can choose to tae nto account the sgnfcance of the covarances Cov(ln y,n+1, ln y j,n+1 j T j for all j > but wth dfferent levels of mpact on the Bermudan swapton s prce. Agan, as there s only one factor n the model we cannot capture everythng. One way s to consder the weghted covarance Cov(ln y,n+1, n j=+1 pt j ln y j,n+1 j T j at each exercse date where the weght s chosen to be a monotoncally decreasng functon n T j p T j = exp[ α(t j ], j >, for some α >. Note that the weghted covarance can be estmated n exactly the same way as we estmate the one step covarance from the maret (see appendx A. As we shall see n secton 4.3, the weghted covarance model spreads the vega responses over the swapton matrx whle the one step covarance model only assgns a sgnfcant contrbuton to the frst column and the reverse dagonal. It s ths feature that gves the weghted covarance model a potental hedgng advantage over the one step covarance model. Model s setup: For = 1,..., n 1, we calbrate the model to the followng maret quantty Cov ma (ln y,n+1, n j=+1 p T j ln y j,n+1 j T j n j=+1 p T j Cov ma (ln y,n+1, ln y j,n+1 j. (12 For ease of exposton, we denote ths maret quantty by B. One can ncorporate the B s nto the model as follows Wthout loss of generalty, fx ξ α T n = σ 2 n,1 T n. At T n 1, we only need to estmate from the maret the one step covarance Cov ma (ln y n 1,2 T n 1, ln y n,1 T n 1 for B n 1 and consequently the one step correlaton Corr ma (ln y n 1,2 T n 1, ln y n,1 T n. Snce we want to calbrate the model s correlaton structure to B n 1, we then requre that ξ α p Tn 1 ξt α σ n 1,2 Tn 1 σ n,1 Tn = B n 1 n ξt α n 1 = Corr ma (ln y n 1,2 T n 1, ln y n,1 T n ξt α n, where we use the relaton n (11. Note that ths step s the same as that n the one step covarance model. Inductvely, assume we are at and have derved {ξ α T j } j=+1,...,n from the prevous steps, ξ α wll then be obtaned by bacward nducton. Snce we have estmated B from the maret and want to calbrate the model s correlaton structure to B, we requre that n j=+1 The superscrpt α s added to emphasze the dependence. p T j ξ α T ξ α T j σ,n+1 T σ j,n+1 j Tj = B, (13 12

13 where we use Corr ma (ln y,n+1, ln y j,n+1 j T j for j > nstead of just j = + 1 as n the one step covarance case. Hence, from (13 we fx ξ α = B σ,n+1 T n j=+1 pt j. (14 1 ξt α σ j,n+1 j Tj j We have now fxed ξ α for = 1,..., n and the SMF model can be mplemented on the grd. One can mmedately see that the one step covarance process s a specal case of the general weghted covarance process when α s very large. The reason s the followng. When α s suffcently large, p T j wll decay exponentally fast and all the weghts wll then become nsgnfcant compared wth p +1. Consequently, only the one step covarance matters n the maret quantty B and the weghted covarance process s reduced to the one step covarance one. Remar 1 For both the one step and weghted covarance models, one can vew the vectors of Blac mpled volatltes ( σ,1, σ,2,..., σ,n+1 for = 1,..., n as the model s ntal nputs (see the global and local fts n appendx A. It follows from the constructons of both the one step and weghted covarance processes that one can wrte ξ α = f (ξ α +1,..., ξ α T n ; { σ, } =1,...,n+1 ; { σ j,n+1 j } j=+1,...,n, (15 where f s some determnstc functon (see appendx A.2 for more detals. Ths cascade structure of the model clearly has an mportant mplcaton on the response of the Bermudan prce to changes n mpled volatltes. Although the one step and weghted covarance processes appear to be more complcated than the HW process, t turns out that they are stll qute smlar n some context. In the next secton, we wll explore further ther connecton through the Bermudan swapton s vegas. 4 Vegas In ths secton, we study the vegas of a Bermudan swapton produced by the dfferent models. Whle the deltas and the gammas do not vary so much from model to model as we shall see n secton 5, the vegas prove to be the most nfluental n the hedgng of a Bermudan. In addton, the underlyng parametrzaton of the model has an mportant mplcaton for the vegas. It s, therefore, worthwhle to nvestgate the behavour of the vegas from dfferent perspectves to explore the model s structure. In subsecton 4.1, we revew analytcally the vegas under dfferent models. We then study the vegas numercally and nvestgate further the ln between them. 4.1 The vega computaton under the swap Marov-functonal model In order to compute the prce of a Bermudan swapton n a SMF model, we need two sets of ntal nputs and a covarance/correlaton structure (captured through the process x. The frst set of nputs are the ntal dscount bonds whch can be safely gnored n ths dscusson as we are only concerned wth the vegas here. The second set of nputs are the co-termnal mpled volatltes whch are used to recover the prces of the underlyng co-termnal vanlla swaptons and fx the functonal forms of the correspondng co-termnal forward swap rates at ther settng dates. In the 13

14 mplementaton of the SMF model, ths s the calbraton to the maret margnals and t s done for all dfferent specfcatons of x. In general, one can vew the value of a Bermudan ˆV T as a prce functon whch maps (the square root of the varances of x and the second set of nputs to a real postve value: ˆV T : R n R n R + ˆV T (ξ, σ := v, (16 where ξ = ( ξ T1, ξ T2,..., ξ Tn and σ = ( σ 1,n, σ 2,n 1,..., σ n,1. For the weghted covarance process, the equvalent nput from x s ξ α = ( ξt α 1, ξt α 2,..., ξt α n. Note that the notaton ξ α when α ndcates the one step covarance case as we explaned n secton For the data we are worng wth, t s observed that the vectors ξ α are qute smlar for all α and hence dfferent choces for α result n smlar prces for the Bermudan. Defne the vega ν, to be the total dervatve of the Bermudan swapton s prce wth respect to σ, for each = 1,..., n and = 1,..., n + 1 : ν, := d ˆV T d σ,. (17 We apply the fnte dfference/bumpng-revaluaton method to calculate these dervatves numercally. We now consder the vegas on a partcular th row of the swapton matrx. In order to dstngush the vegas produced by dfferent models, we denote ν, α for the weghted covarance process. Note agan that the notaton ν, α as α ndcates the vegas for the one step covarance process. For the HW and MR models, we denote the vegas by ν, hw and νmr, respectvely. Note that only the co-termnal vegas ν,n+1 hw and νmr,n+1 matter n the HW and MR models whch follows drectly from ther setups. The other vegas are all zero for these models. 1. = 1,..., n (off reverse dagonal: For the one step and weghted covarance models, as σ, s only nvolved n ξ α, by the chan rule and equaton (15 we have that ν α, = d ˆV T d σ, = s=1 ˆV T ξ α T s d ξ α Ts. (18 d σ, The term ˆV T ξ α Ts should be nterpreted as the partal dervatve of the prce functon ˆV T wth respect to the s th coordnate of the vector ξ α. For the total dervatves d ξ α Ts d σ, where 1 s <, by equaton (15 t s clear that the dependence of ξ α T s on σ, s through { ξ α T j } j =s+1,...,. Note that the prce of a Bermudan swapton s not senstve to these mpled volatlty nputs under the HW and MR models. Hence, as noted above we have zero values for the correspondng vegas n these models. 2. = n + 1 (reverse dagonal: For the one step and weghted covarance models, as σ,n+1 s nvolved n both ξ α and σ we have that ν,n+1 α = ˆV T + ˆV T σ,n+1 ξt α d ξ α Ts. (19 s d σ,n+1 14 s=1

15 Note that the frst term on the rght hand sde of (19 s approxmately the th bucet vega ν,n+1 mr of the MR process when the prces are comparable between models. Ths term only reflects the change of the margnal dstrbuton of y,n+1 under ts own swapton measure. For the HW process, the equvalent vega s ν hw,n+1 = ˆV T + ˆV T σ,n+1 d ξt (2 ξ T d σ,n+1 ξt ν,n+1 mr + ˆV T d. ξ T d σ,n+1 One can see that the above dfferences between ν,n+1 hw and να,n+1 on one hand and νhw,n+1 and ν,n+1 mr on the other hand come from the dfferences n ther correlaton structures. Snce the HW and the one step and the weghted covarance models are dfferent examples of parametrzaton by tme, ther vegas are closely connected as we shall see n secton 4.4. For the MR and HW models whch respond to the reverse dagonal only, the dfference n ther correlaton structures follow drectly from the dfference between the parameterzatons by expry and by tme. We wll revew ths dfference n the next secton. 4.2 The Bermudan swapton s vegas under the HW and MR models The example we consder here s a 11 years annual Bermudan swapton wth fxed rate K = 5%, notonal N = 1 mllon and the followng ntal data: Tenor Expry and Table 4.1: Blac mpled volatltes (% of the ATM swaptons on October 17,

16 Tenor Expry Table 4.2: Intal swap rates (% on October 17, 27. Here we compare the vegas of the Bermudan swapton under the HW and MR models. The mean reverson parameter a s fxed at 3% for both two drvng processes so that the Bermudan prces produced by the two models are close and also comparable to those that are produced by the one step and weghted covarance models. We dsplay the vegas n tables 4.3 and 4.4. The poston of each vega corresponds to the mpled volatlty n the swapton matrx. Recall that the off reverse dagonal entres are all zero snce the Bermudan swapton s prce s not senstve to the correspondng mpled volatlty nputs here. Remar 2 In practce, traders usually quote vega as the change n prce when mpled volatlty ncreases by 1 bass ponts (bp or 1% so we wll scale the true vega by a factor of.1,.e. ν,.1ν,. For example, the entry 4.9 n the frst row and the last column of table 4.3 means that when σ 1,1 ncreases by 1% the Bermudan prce (wth notonal 1 mllon wll ncrease by 4, 9. Tenor Expry Table 4.3: The Bermudan swapton s scaled vegas (n 1 4 under the HW model. 16

17 Tenor Expry Table 4.4: The Bermudan swapton s scaled vegas (n 1 4 under the MR model. It s seen n fgure 4.1 that the vegas for both models as a functon of expry dsplay humped shapes whose peas are attaned at the same exercse date. We also observe that the HW vegas are lower than the MR vegas at the early exercse dates but hgher at the later ones. Recall from subsecton 4.1 that ths dfference n vegas s actually caused by the dfference n the correlaton structures. Ths s rather mportant for a strongly correlaton-dependent product le the Bermudan swapton. In the followng, we wll gve a crude explanaton on how a change n one of the cotermnal mpled volatltes can affect the correlatons of the co-termnal swap rates under the two models whch wll then clearly ndcate ther vegas. Parametrzaton by expry (MR: We recall that the MR process s unaltered by bumpng any co-termnal mpled volatlty. Ths then mples that the correlaton structure of the model/co-termnal forward swap rates s unaltered,.e. Corr mo (ln y,n+1, ln y j,n+1 j whch ξmn(t,t j T j are approxmately ξ max(t,t j are unchanged for all, j n. Parametrzaton by tme (HW: Bumpng σ,n+1 has an mmedate effect on the HW process and hence the correlaton structure of the model/co-termnal forward swap rates. Specfcally, ξmn(t,t j Corr mo (ln y,n+1, ln y j,n+1 j T j ξ wll ncrease for j > but decrease for j <. max(t,t j Heurstcally, we have the followng overall effect: on average the co-termnal forward swap rates wll tend to be more correlated f s small and less correlated f s large. We employ the followng heurstc argument for the correlaton s effects on the Bermudan prce. The optonalty of a Bermudan mples that the lower the correlatons of the co-termnal swap rates get, the hgher the prce of the Bermudan swapton becomes. For more detals of how correlatons affect the Bermudan prce, see (Andersen & Pterbarg, 21 and (Rebonato, 24 for example for reference. Hence, we draw the followng concluson on the vegas. If x s parameterzed by tme and s small (early exercse date, the co-termnal forward swap rates wll tend to be more correlated on average. Ths effect wll cause the HW prce to ncrease less than the MR prce and mae ν,n+1 hw lower than ν,n+1 mr. On the other hand, f s large (late exercse date, the co-termnal forward swap rates wll tend to be less correlated on average. Ths then causes ν,n+1 hw to be hgher than ν,n+1 mr. Ths fundamental dfference s the ey observaton whch leads to very dfferent hedgng profles as we shall see later. 17

18 Remar 3 The MR vegas become very small or even negatve at the end of the opton whch s possble under some crcumstances n practce. See appendx B n (Petersz & Pelsser, 24 for an explanaton of negatve vega for a two stoc Bermudan opton example. 4.3 The Bermudan swapton s vegas under the one step and weghted covarance models We test the one step and weghted covarance processes wth dfferent values of α and dsplay ther vega matrces n tables 4.5, 4.6, 4.7 and 4.8. We frst loo at the vegas for the one step covarance model n table 4.5. The frst thng to notce from ths table s that the vega response starts shftng away from the reverse dagonal entres. We obtan a vega profle whch assgns a sgnfcant contrbuton to the frst column of the swapton matrx. The other vega entres are seen to be much smaller and very close to zero except for the co-termnal ones. The vega behavour of the frst column can be seen from the local ft n appendx A.2. In ths local ft step, apart from the reverse dagonal entres we see that shftng the frst column has the most dstnctve effect on the one step covarance terms that we estmate from the maret. The one step covarance model s set up such that t responds to the changes n the one step covarances only (not the other covarances as consdered n the weghted covarance model. Thus t s a shft n the frst column or a reverse dagonal entry of the swapton matrx that has the largest vega response. Note that the swaptons correspondng to the frst column are farly llqud so t would be desrable to use more mpled volatltes to moderate the response to any naccurate maret sgnals. The results produced by the weghted covarance model ndeed have ths feature. For the weghted covarance model, we observe dfferent patterns for the vega response dependng on dfferent values of α. For example, when α =.5 we observe a bgger response n the central part of the table compared wth that when we use a much hgher value of α, for nstance α = 5. For some partcular rows, t s seen that those central entres even domnate the reverse dagonal and the frst column. For larger values of α such as.3 n table 4.7, we observe a clearer trend n the vega entres. They tend to ncrease n tenor for each expry. When α gets much hgher (table 4.8, t s clear that the entres loo very smlar to the one step covarance case where the vegas from the central part become much more nsgnfcant and domnated by the reverse dagonal and the frst column. Ths s predctable as one step covarance s a specal case of the weghted covarance model when α s very large. Remar 4 We get a few negatve vega entres when α =.5. Note that those n the reverse dagonal (the frst two rows are qute large n magntude. One reason for ths behavour s the followng. When we shft the co-termnal mpled volatlty σ,n+1, the approxmatons from the maret for the correlatons Corr ma (ln y,n+1, ln y j,n+1 j T j tend to ncrease for j >. For a very low value of α, the model wll tae nto account all these ncreases n correlatons wth hgh levels of mpact on the Bermudan prce snce the geometrc weght p T j decays very slowly n T j. Therefore, when s small the overall ncrease n correlatons of the co-termnal swap rates could be large whch n turn leads to a decrease n prce and hence negatve vegas. 18

19 Tenor Expry Table 4.5: The Bermudan swapton s scaled vegas (n 1 4 under the one step covarance model. Tenor Expry Table 4.6: The Bermudan swapton s scaled vegas (n 1 4 under the weghted covarance model (α =.5. 19

20 Tenor Expry Table 4.7: The Bermudan swapton s scaled vegas (n 1 4 under the weghted covarance model (α =.3. Tenor Expry Table 4.8: The Bermudan swapton s scaled vegas (n 1 4 under the weghted covarance model (α = The net maret vegas for dfferent parameterzatons We recall that the HW process s an example of parametrzaton by tme and t has a certan vega profle wth the responses only on the reverse dagonal. The one step and weghted covarance models move the vega response away from the reverse dagonal and ths causes ther hedgng behavours to be qute dfferent from that of the HW model. However, ther vega profles are stll very closely connected as they are dfferent examples of parametrzaton by tme. For the one step and weghted covarance models, we plot the sum of the vegas for each row (expry of the swapton matrx. Wth our ntal data, we observe that each row sum s roughly a constant that s ndependent of α and very close to the co-termnal vega on the same row of the HW model (fgure

21 Net scaled vega (row sum 16 Vega Comparson HW MR α =.5 α =.3 α = 5 One step cov Row Fgure 4.1: The (net row sum of the scaled vegas (n 1 4 of a 11-years annual Bermudan swapton for dfferent models and parameters. We state ths observaton as a result. Result 1 For each = 1,..., n and all α >, under the assumptons that mpled volatltes of the same expry are not so varant wth respect to tenor and the varances ξ T1,..., ξ Tn are comparable between models the followng relaton holds true n+1 =1 ν α, νhw,n+1 (21 In order to prove ths result, we need the followng sub-result that wors wth the log-transformaton of the mpled volatltes. For = 1,..., n + 1, let Σ, := ln σ,. We defne the total dervatve of the Bermudan swapton s prce wth respect to these log-mpled volatltes as ˆν, := d ˆV T dσ,. Agan, n order to dstngush dfferent models we denote ˆν, α for the one step and weghted covarance models. For the HW model, the equvalent co-termnal term s denoted by ˆν,n+1 hw and note that t s the only term that matters. We state the sub-result as a lemma. Lemma 1 For each = 1,..., n and all α >, under the second assumpton n result 1 the followng relaton holds true n+1 =1 ˆν α, ˆνhw,n+1 (22 21

22 Proof: We wll frst prove that for the one step and weghted covarance models the row sum n+1 =1 ˆν, α s ndependent of α. Note that n+1 =1 ˆν, α roughly represents the effect of a parallel addtve shft of Σ, for = 1,..., n + 1 on the Bermudan prce as we can see from the Taylor expanson of the prce functon ˆV T. Ths s equvalent to a parallel multplcatve shft of σ, for = 1,..., n + 1 snce Σ, Σ, + ln ɛ σ, ɛ σ,. One then can wrte n+1 =1 ˆν, α ˆV T (ɛ σ,1,..., ɛ σ,n+1 ˆV T ( σ,1,..., σ,n+1, ln ɛ where ɛ > 1 and suffcently small. It remans to show that the effect of the parallel multplcatve shft of the th row of the swapton matrx on the Bermudan prce s ndependent of α. In the followng we use the analyss obtaned n appendx A. The man purpose s to assess the effects of the above parallel multplcatve shft on each of the estmated covarances that feed nto the one step and weghted covarance models,.e. Cov ma (ln y,n+1, ln y j,n+1 j for j = +1,..., n. Ths can be seen va the effects on the covarances of the log-libors. From the local ft n appendx A.2, we have the followng approxmaton for the covarances of the log-libors at each exercse date σ 2, + 1 l= + 1 l = ζ, l (ζ, l (Cov ma (ln L l, ln L l, = 1,..., n + 1, where {ζ, l (} l=,...,n are constants that only depend on the ntal dscount curve (see appendx A. Therefore, under a parallel multplcatve shft of the th row of the swapton matrx we have that Cov ma (ln L l, ln LT l ɛ 2 Cov ma (ln L l, ln L l, l, l =,..., n. Snce the covarance of the log-swap rates can be approxmated by summng up the covarances of the correspondng spannng log-libors (see appendx A.2, we have Cov ma (ln y,n+1, ln y j,n+1 j n n = l=j ζ,n+1 (ζ j,n+1 j l (Cov ma (ln L, ln L l Cov ma (ln y,n+1, ln y j,n+1 j ɛ 2 Cov ma (ln y,n+1, ln y j,n+1 j, j = + 1,..., n, Recall the maret quantty B = n j=+1 pt j Cov ma (ln y,n+1, ln y j,n+1 j. For all values of α, we then have that B ɛ 2 B. We further recall the constructon of ξt α n equaton (14 where we have ξ α = B σ,n+1 T n j=+1 pt j. 1 ξt α σ j,n+1 j Tj j 22

23 Smlar to the arguments n secton 4.1, the cascade structure n equaton (15 mples that { ξ α T j } j=+1,...,n are nvarant under a parallel multplcatve shft of the th row. It then follows from (14 that ξt α ɛ ξt α, and hence further 1 ξ α σ,n+1 T s nvarant. For 1 s <, we have that ξ α T s = B s σ s,n+1 s Ts n j=s+1 pt j T s. 1 ξt α σ j,n+1 j Tj j When s = 1, t s clear that ξt α s s also nvarant because 1 ξt α σ,n+1 T s nvarant. Inductvely, we have that { ξt α s } 1 s< are all nvarant. It s now clear that a parallel multplcatve shft of the th row wll only shft ξt α ɛ ξt α regardless of α. Snce Σ, s just the log of σ, for each = 1,..., n + 1, one can wrte the analogous formulae for ˆν, α followng exactly the same arguments as n (18 and (19. Therefore, for the off reverse dagonal entres: = 1,..., n ˆν, α = ˆV T ξt α s s=1 d ξ α Ts, dσ, and for the reverse dagonal entry = n + 1 Hence, we have the row sum ˆν α,n+1 = ˆV T Σ,n+1 + ˆV T ξt α d ξ α Ts. s dσ,n+1 s=1 n+1 =1 ˆν α, = = ˆV n+1 T + Σ,n+1 ˆV T Σ,n+1 + =1 s=1 s=1 ˆV T ξ α T s ˆV T ξ α T s d ξ α Ts dσ, ( n+1 =1 d ξ α T s dσ,. Smlar to that we dscussed earler, the sum n+1 =1 d ξ α Ts dσ, roughly represents the effect of a parallel multplcatve shft of the th row of the swapton matrx on ξ α T s by loong at the Taylor expanson on ξ α T s. We prevously concluded for the one step and weghted covarance models that ths shft wll leave ξ α T j unchanged for j. It follows that n+1 =1 n+1 =1 ˆν α, ˆV T + ˆV T Σ,n+1 n+1 ξ α T =1 d ξ α Ts dσ, d ξt α. dσ, for s <. Hence 23